This is a first revision of the fifth competitive renewal of a grant whose purpose is to conduct applied statistical research motivated by epidemiological studies related to carcinogenicity and adverse birth and reproductive outcomes. Studies designed to assess the health impact of environmental exposures are the primary emphasis. Specific Aims are: 1. To develop, implement and evaluate strategies for efficient sub-sampling from repeated measures and longitudinal studies where resource limitations preclude measuring exposures of interest on all study participants at all follow-up times. In particular, a. To develop, extend and apply longitudinal case control study designs for longitudinal birth cohorts and to use these methods as the basis for optimal designs that involve subsampling from a longitudinal cohort;b. To develop and evaluate an efficient design and analysis for a reproductive toxicology study wherein pregnancy may fail at any of a number of different points in gestation. 2. To develop and evaluate a broad class of penalized latent variable models for the analysis of high dimensional exposure and response data. In particular, a. To develop penalized item response theory (IRT) models and use them to characterize the relationship between a moderate number of related binary exposures and a binary response variable;b. To extend penalized latent class models to incorporate exposure/response relationships, including gene-environment interactions, in studies involving a moderate number of genetic markers;and c. To extend the methods developed in Aims 2a and 2b to apply in truly high dimensional settings. 3. To extend, evaluate and apply methods for assessing goodness-of-fit for correlated data and latent variable models via the use of Cholesky residuals.